3.116 \(\int \frac{1}{x^3 (a^2+2 a b x^3+b^2 x^6)^{5/2}} \, dx\)

Optimal. Leaf size=398 \[ -\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac{7}{54 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}-\frac{770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{770 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

[Out]

154/(243*a^4*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x^2*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
+ 7/(54*a^2*x^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 77/(324*a^3*x^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6]) - (385*(a + b*x^3))/(243*a^5*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (770*b^(2/3)*(a + b*x^3)*
ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7
70*b^(2/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(729*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (385*b^(2/3)
*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(729*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

________________________________________________________________________________________

Rubi [A]  time = 0.210628, antiderivative size = 398, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.346, Rules used = {1355, 290, 325, 200, 31, 634, 617, 204, 628} \[ -\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )}+\frac{7}{54 a^2 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^2}+\frac{1}{12 a x^2 \sqrt{a^2+2 a b x^3+b^2 x^6} \left (a+b x^3\right )^3}-\frac{770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{770 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

154/(243*a^4*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 1/(12*a*x^2*(a + b*x^3)^3*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])
+ 7/(54*a^2*x^2*(a + b*x^3)^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + 77/(324*a^3*x^2*(a + b*x^3)*Sqrt[a^2 + 2*a*b
*x^3 + b^2*x^6]) - (385*(a + b*x^3))/(243*a^5*x^2*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (770*b^(2/3)*(a + b*x^3)*
ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(243*Sqrt[3]*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) - (7
70*b^(2/3)*(a + b*x^3)*Log[a^(1/3) + b^(1/3)*x])/(729*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6]) + (385*b^(2/3)
*(a + b*x^3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(729*a^(17/3)*Sqrt[a^2 + 2*a*b*x^3 + b^2*x^6])

Rule 1355

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_), x_Symbol] :> Dist[(a + b*x^n + c*x^
(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*FracPart[p])), Int[(d*x)^m*(b/2 + c*x^n)^(2*p), x], x] /; Fr
eeQ[{a, b, c, d, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2]

Rule 290

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m}, x] && IGtQ[n, 0] && LtQ[p, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 200

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Dist[1/(3*Rt[a, 3]^2), Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Di
st[1/(3*Rt[a, 3]^2), Int[(2*Rt[a, 3] - Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x]
/; FreeQ[{a, b}, x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{1}{x^3 \left (a^2+2 a b x^3+b^2 x^6\right )^{5/2}} \, dx &=\frac{\left (b^4 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x^3\right )^5} \, dx}{\sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (7 b^3 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x^3\right )^4} \, dx}{6 a \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (77 b^2 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x^3\right )^3} \, dx}{54 a^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (154 b \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x^3\right )^2} \, dx}{81 a^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (770 \left (a b+b^2 x^3\right )\right ) \int \frac{1}{x^3 \left (a b+b^2 x^3\right )} \, dx}{243 a^4 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (770 b \left (a b+b^2 x^3\right )\right ) \int \frac{1}{a b+b^2 x^3} \, dx}{243 a^5 \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (770 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac{1}{\sqrt [3]{a} \sqrt [3]{b}+b^{2/3} x} \, dx}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (770 \sqrt [3]{b} \left (a b+b^2 x^3\right )\right ) \int \frac{2 \sqrt [3]{a} \sqrt [3]{b}-b^{2/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{\left (385 \left (a b+b^2 x^3\right )\right ) \int \frac{-\sqrt [3]{a} b+2 b^{4/3} x}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{729 a^{17/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (385 b^{2/3} \left (a b+b^2 x^3\right )\right ) \int \frac{1}{a^{2/3} b^{2/3}-\sqrt [3]{a} b x+b^{4/3} x^2} \, dx}{243 a^{16/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{\left (770 \left (a b+b^2 x^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{243 a^{17/3} \sqrt [3]{b} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ &=\frac{154}{243 a^4 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{1}{12 a x^2 \left (a+b x^3\right )^3 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{7}{54 a^2 x^2 \left (a+b x^3\right )^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{77}{324 a^3 x^2 \left (a+b x^3\right ) \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{385 \left (a+b x^3\right )}{243 a^5 x^2 \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{770 b^{2/3} \left (a+b x^3\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{243 \sqrt{3} a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}-\frac{770 b^{2/3} \left (a+b x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}+\frac{385 b^{2/3} \left (a+b x^3\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{729 a^{17/3} \sqrt{a^2+2 a b x^3+b^2 x^6}}\\ \end{align*}

Mathematica [A]  time = 0.135487, size = 234, normalized size = 0.59 \[ \frac{\left (a+b x^3\right ) \left (1540 b^{2/3} \left (a+b x^3\right )^4 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-\frac{1458 a^{2/3} \left (a+b x^3\right )^4}{x^2}-3162 a^{2/3} b x \left (a+b x^3\right )^3-1314 a^{5/3} b x \left (a+b x^3\right )^2-621 a^{8/3} b x \left (a+b x^3\right )-243 a^{11/3} b x-3080 b^{2/3} \left (a+b x^3\right )^4 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-3080 \sqrt{3} b^{2/3} \left (a+b x^3\right )^4 \tan ^{-1}\left (\frac{2 \sqrt [3]{b} x-\sqrt [3]{a}}{\sqrt{3} \sqrt [3]{a}}\right )\right )}{2916 a^{17/3} \left (\left (a+b x^3\right )^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(a^2 + 2*a*b*x^3 + b^2*x^6)^(5/2)),x]

[Out]

((a + b*x^3)*(-243*a^(11/3)*b*x - 621*a^(8/3)*b*x*(a + b*x^3) - 1314*a^(5/3)*b*x*(a + b*x^3)^2 - 3162*a^(2/3)*
b*x*(a + b*x^3)^3 - (1458*a^(2/3)*(a + b*x^3)^4)/x^2 - 3080*Sqrt[3]*b^(2/3)*(a + b*x^3)^4*ArcTan[(-a^(1/3) + 2
*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] - 3080*b^(2/3)*(a + b*x^3)^4*Log[a^(1/3) + b^(1/3)*x] + 1540*b^(2/3)*(a + b*x^3
)^4*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]))/(2916*a^(17/3)*((a + b*x^3)^2)^(5/2))

________________________________________________________________________________________

Maple [B]  time = 0.022, size = 542, normalized size = 1.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x)

[Out]

-1/2916*(-3080*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^14*b^4+3080*ln(x+(a/b)^(1/3))*x^14
*b^4-1540*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^14*b^4+4620*(a/b)^(2/3)*x^12*b^4-12320*3^(1/2)*arctan(1/3*3^(1/2
)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^11*a*b^3+12320*ln(x+(a/b)^(1/3))*x^11*a*b^3-6160*ln(x^2-(a/b)^(1/3)*x+(a/b
)^(2/3))*x^11*a*b^3+16632*(a/b)^(2/3)*x^9*a*b^3-18480*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3
))*x^8*a^2*b^2+18480*ln(x+(a/b)^(1/3))*x^8*a^2*b^2-9240*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^8*a^2*b^2+21483*(a
/b)^(2/3)*x^6*a^2*b^2-12320*3^(1/2)*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^5*a^3*b+12320*ln(x+(a
/b)^(1/3))*x^5*a^3*b-6160*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))*x^5*a^3*b+11172*(a/b)^(2/3)*x^3*a^3*b-3080*3^(1/2)
*arctan(1/3*3^(1/2)*(-2*x+(a/b)^(1/3))/(a/b)^(1/3))*x^2*a^4+3080*ln(x+(a/b)^(1/3))*x^2*a^4-1540*ln(x^2-(a/b)^(
1/3)*x+(a/b)^(2/3))*x^2*a^4+1458*(a/b)^(2/3)*a^4)*(b*x^3+a)/(a/b)^(2/3)/x^2/a^5/((b*x^3+a)^2)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.57481, size = 799, normalized size = 2.01 \begin{align*} -\frac{4620 \, b^{4} x^{12} + 16632 \, a b^{3} x^{9} + 21483 \, a^{2} b^{2} x^{6} + 11172 \, a^{3} b x^{3} + 1458 \, a^{4} - 3080 \, \sqrt{3}{\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \arctan \left (\frac{2 \, \sqrt{3} a x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}} - \sqrt{3} b}{3 \, b}\right ) + 1540 \,{\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b^{2} x^{2} + a b x \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} + a^{2} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{2}{3}}\right ) - 3080 \,{\left (b^{4} x^{14} + 4 \, a b^{3} x^{11} + 6 \, a^{2} b^{2} x^{8} + 4 \, a^{3} b x^{5} + a^{4} x^{2}\right )} \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}} \log \left (b x - a \left (-\frac{b^{2}}{a^{2}}\right )^{\frac{1}{3}}\right )}{2916 \,{\left (a^{5} b^{4} x^{14} + 4 \, a^{6} b^{3} x^{11} + 6 \, a^{7} b^{2} x^{8} + 4 \, a^{8} b x^{5} + a^{9} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="fricas")

[Out]

-1/2916*(4620*b^4*x^12 + 16632*a*b^3*x^9 + 21483*a^2*b^2*x^6 + 11172*a^3*b*x^3 + 1458*a^4 - 3080*sqrt(3)*(b^4*
x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(-b^2/
a^2)^(2/3) - sqrt(3)*b)/b) + 1540*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)
^(1/3)*log(b^2*x^2 + a*b*x*(-b^2/a^2)^(1/3) + a^2*(-b^2/a^2)^(2/3)) - 3080*(b^4*x^14 + 4*a*b^3*x^11 + 6*a^2*b^
2*x^8 + 4*a^3*b*x^5 + a^4*x^2)*(-b^2/a^2)^(1/3)*log(b*x - a*(-b^2/a^2)^(1/3)))/(a^5*b^4*x^14 + 4*a^6*b^3*x^11
+ 6*a^7*b^2*x^8 + 4*a^8*b*x^5 + a^9*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x^{3} \left (\left (a + b x^{3}\right )^{2}\right )^{\frac{5}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**3/(b**2*x**6+2*a*b*x**3+a**2)**(5/2),x)

[Out]

Integral(1/(x**3*((a + b*x**3)**2)**(5/2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(b^2*x^6+2*a*b*x^3+a^2)^(5/2),x, algorithm="giac")

[Out]

sage0*x